It is proved that if $F$ is an infinite field with characteristic different from $2$</formtex>, whose theory is supersimple, and $C$</formtex> is an elliptic or hyperelliptic curve over $F$</formtex> with generic modulus, then $C$</formtex> has a generic $F$</formtex>-rational point. The notion of generity here is in the sense of the supersimple field $F$</formtex>.